Convergence of sequences of iterates of random-valued vector functions

Rafał Kapica¹

¹ Institute of Mathematics Silesian University Bankowa 14 40-007 Katowice, Poland

Colloquium Mathematicum, Tome 97 (2003) no. 1, pp. 1-6.

Résumé

Given a probability space $({\mit \Omega },{{\mathcal A}},P)$ and a closed subset $X$ of a Banach lattice, we consider functions $f:X\times {\mit \Omega }\to X$ and their iterates $f^n:X\times {\mit \Omega }^{{{\mathbb N}}}\to X$ defined by $f^1(x,\omega )=f(x,\omega _1)$, $f^{n+1}(x,\omega )=f(f^n(x,\omega ),\omega _{n+1})$, and obtain theorems on the convergence (a.s. and in $L^1$) of the sequence $(f^n(x,\cdot ))$.

Détail

DOI : 10.4064/cm97-1-1

Keywords: given probability space mit omega mathcal closed subset banach lattice consider functions times mit omega their iterates times mit omega mathbb defined omega omega omega x omega omega obtain theorems convergence sequence cdot